\(\int \frac {\sqrt {c+d x^2} (e+f x^2)}{\sqrt {a+b x^2}} \, dx\) [22]

Optimal result

Integrand size = 30, antiderivative size = 281 \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {(3 b d e+b c f-2 a d f) x \sqrt {c+d x^2}}{3 b d \sqrt {a+b x^2}}+\frac {f x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}-\frac {\sqrt {a} (3 b d e+b c f-2 a d f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 b^{3/2} d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} (3 b e-a f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 b^{3/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

1/3*(-2*a*d*f+b*c*f+3*b*d*e)*x*(d*x^2+c)^(1/2)/b/d/(b*x^2+a)^(1/2)+1/3*f*x 
*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b-1/3*a^(1/2)*(-2*a*d*f+b*c*f+3*b*d*e)*(d 
*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1 
/2))/b^(3/2)/d/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/3*a^(1/2) 
*(-a*f+3*b*e)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1 
-a*d/b/c)^(1/2))/b^(3/2)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.91 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {i c (2 a d f-b (3 d e+c f)) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+f \left (\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right )-i c (-b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b \sqrt {\frac {b}{a}} d \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(Sqrt[c + d*x^2]*(e + f*x^2))/Sqrt[a + b*x^2],x]
 

Output:

(I*c*(2*a*d*f - b*(3*d*e + c*f))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*E 
llipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + f*(Sqrt[b/a]*d*x*(a + b*x^ 
2)*(c + d*x^2) - I*c*(-(b*c) + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c 
]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*b*Sqrt[b/a]*d*Sqrt[a 
 + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {403, 406, 320, 388, 313}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\int \frac {(3 b d e+b c f-2 a d f) x^2+c (3 b e-a f)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}+\frac {f x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {c (3 b e-a f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+(-2 a d f+b c f+3 b d e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}+\frac {f x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {(-2 a d f+b c f+3 b d e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (3 b e-a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b}+\frac {f x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {(-2 a d f+b c f+3 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (3 b e-a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b}+\frac {f x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\frac {c^{3/2} \sqrt {a+b x^2} (3 b e-a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(-2 a d f+b c f+3 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 b}+\frac {f x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}\)

Input:

Int[(Sqrt[c + d*x^2]*(e + f*x^2))/Sqrt[a + b*x^2],x]
 

Output:

(f*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*b) + ((3*b*d*e + b*c*f - 2*a*d*f) 
*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*Ellip 
ticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a 
+ b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (c^(3/2)*(3*b*e - a*f)*Sqrt 
[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sq 
rt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*b)
 

Defintions of rubi rules used

rule 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

rule 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

Maple [A] (verified)

Time = 6.78 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.11

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {f x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b}+\frac {\left (c e -\frac {a c f}{3 b}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (c f +d e -\frac {f \left (2 a d +2 b c \right )}{3 b}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(312\)
risch	\(\frac {f x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 b}-\frac {\left (-\frac {\left (2 a d f -b c f -3 b d e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {a c f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {3 b c e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 b \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(386\)
default	\(\frac {\sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}\, \left (\sqrt {-\frac {b}{a}}\, b \,d^{2} f \,x^{5}+\sqrt {-\frac {b}{a}}\, a \,d^{2} f \,x^{3}+\sqrt {-\frac {b}{a}}\, b c d f \,x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d f -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} f -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d f +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} f +3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d e +\sqrt {-\frac {b}{a}}\, a c d f x \right )}{3 \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) b \sqrt {-\frac {b}{a}}\, d}\)	\(394\)

Input:

int((d*x^2+c)^(1/2)*(f*x^2+e)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/3*f/b*x*(b* 
d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+(c*e-1/3*a*c*f/b)/(-b/a)^(1/2)*(1+b*x^2/a 
)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x* 
(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(c*f+d*e-1/3*f/b*(2*a*d+2*b*c))*c/( 
-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a 
*c)^(1/2)/d*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE( 
x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=-\frac {{\left (3 \, b c d e + {\left (b c^{2} - 2 \, a c d\right )} f\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d} {\left (3 \, {\left (b c d + b d^{2}\right )} e + {\left (b c^{2} - 2 \, a c d - a d^{2}\right )} f\right )} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (b d^{2} f x^{2} + 3 \, b d^{2} e + {\left (b c d - 2 \, a d^{2}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, b^{2} d^{2} x} \] Input:

integrate((d*x^2+c)^(1/2)*(f*x^2+e)/(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

-1/3*((3*b*c*d*e + (b*c^2 - 2*a*c*d)*f)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e( 
arcsin(sqrt(-c/d)/x), a*d/(b*c)) - sqrt(b*d)*(3*(b*c*d + b*d^2)*e + (b*c^2 
 - 2*a*c*d - a*d^2)*f)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/( 
b*c)) - (b*d^2*f*x^2 + 3*b*d^2*e + (b*c*d - 2*a*d^2)*f)*sqrt(b*x^2 + a)*sq 
rt(d*x^2 + c))/(b^2*d^2*x)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\int \frac {\sqrt {c + d x^{2}} \left (e + f x^{2}\right )}{\sqrt {a + b x^{2}}}\, dx \] Input:

integrate((d*x**2+c)**(1/2)*(f*x**2+e)/(b*x**2+a)**(1/2),x)
 

Output:

Integral(sqrt(c + d*x**2)*(e + f*x**2)/sqrt(a + b*x**2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\int { \frac {\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}}{\sqrt {b x^{2} + a}} \,d x } \] Input:

integrate((d*x^2+c)^(1/2)*(f*x^2+e)/(b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(d*x^2 + c)*(f*x^2 + e)/sqrt(b*x^2 + a), x)
 

Giac [F]

\[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\int { \frac {\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}}{\sqrt {b x^{2} + a}} \,d x } \] Input:

integrate((d*x^2+c)^(1/2)*(f*x^2+e)/(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(d*x^2 + c)*(f*x^2 + e)/sqrt(b*x^2 + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\int \frac {\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )}{\sqrt {b\,x^2+a}} \,d x \] Input:

int(((c + d*x^2)^(1/2)*(e + f*x^2))/(a + b*x^2)^(1/2),x)
 

Output:

int(((c + d*x^2)^(1/2)*(e + f*x^2))/(a + b*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, f x -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a d f +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b c f +3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b d e -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a c f +3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b c e}{3 b} \] Input:

int((d*x^2+c)^(1/2)*(f*x^2+e)/(b*x^2+a)^(1/2),x)
 

Output:

(sqrt(c + d*x**2)*sqrt(a + b*x**2)*f*x - 2*int((sqrt(c + d*x**2)*sqrt(a + 
b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*d*f + int((sqrt( 
c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4), 
x)*b*c*f + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 
+ b*c*x**2 + b*d*x**4),x)*b*d*e - int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/ 
(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*c*f + 3*int((sqrt(c + d*x**2)* 
sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*b*c*e)/(3*b)